Power blackouts occur around the world with more frequency as the demand for additional power continues to increase throughout the industrial societies. As each geographic area develops a need for electricity, a new power plant or power distribution system must be constructed. Each power generating system is unique, because each comprises different grids of power-generating capacity that are strung together to meet the varying power requirements of a given populace and geographic region.
Generally speaking, the power-generating capacities of most systems throughout the world are not keeping pace with the demand. This situation requires corrective stopgap measures, such as the institution of "brownout" procedures during periods of peak power loads. Brownout procedures can entail the reduction of voltage, as well as the number of cycles being supplied each second. Such procedures can only be a temporary means to alleviate an increased power demand problem. Eventually the system will fail, unless the generating power capacity is increased. Enforcing such curtailments, however, is not an adequate answer to the power shortage problem. Certain equipment, such as sensitive electronic devices and computers, will not run properly under reduced voltage and/or reduced cycle conditions.
Aging power facilities are a critical problem. Many power plants are currently in need of repair and/or are seriously inefficient. Many power generation facilities (such as those producing nuclear power) are dangerously outmoded and must be taken off line. Therefore, many of the existent power-generating systems are actually shrinking in their capacity to supply power.
Due to economics and the sometimes inordinate amount of time inherent in planning, construction of new generating plants to increase power capacity is not always possible. Even if power-generating capacity could be enhanced, it has not been best determined just where in the system the added capacity would provide the most benefit. In other words, there is no present means for determining the so-called "weak areas" in each power-generating system. Despite stopgap measures and the addition of new generating capacity, many systems still fail prey to blackouts. Present-day blackouts result from the industry's inability to predict and correct for these weak areas.
Weak areas are defined as those parts of the grid of a power-generating system that cannot tolerate added load. For example, present-day calculations cannot provide answers as to whether a particular bus can withstand a given increase in load, or whether the system can withstand a simultaneous increase of load on one bus while sustaining an increase in demand on another bus in the grid.
Voltage collapse is generally caused by two types of system disturbances: load variations and contingencies. The voltage collapse in Japan in 1987 was due to load variations; the load collapse in Sweden in 1982 was due to a contingency. Blackouts usually develop in systems that are heavily loaded and which also experience additional power demand.
The present invention has developed a method for determining a performance index for each power-generating system that, for the first time, is directly correlated to load demand. This performance index call be easily interpreted by power engineers and operators, so that weak areas and potential voltage collapse conditions can be predicted and corrected.
When the underlying cause of collapse is due to load variations, the performance index of the current invention has the ability to measure the amount of load increase that the system can tolerate prior to such collapse.
When the underlying collapse mechanism is due to a contingency, the performance index of this invention can measure its severity. The performance index of this invention can also assess whether the system can sustain a contingency without collapse.
The method of this invention uses the new performance index to provide a means by which load-shedding techniques can be optimized to prevent system collapse.
The method of voltage collapse prevention of the invention comprises the following general steps:
a) measuring the load demands in the power-generating system; PA1 b) calculating a voltage collapse index (VCI) directly correlated to the load demands of the power-generating system of step (a); PA1 c) utilizing the VCI to identify weak areas in the grid of the power-generating system; PA1 d) determining actual safety margins that exist for these weak areas before collapse will occur; and PA1 e) shedding power in those portions of the grid that are identified as weak areas, in order to prevent voltage collapse in the power-generating system, in accordance with said safety margins. PA1 a) measuring the load demands in the power-generating system; PA1 b) calculating a voltage collapse index (VCI) directly correlated to the load demands of the power-generating system of step (a); PA1 c) utilizing the VCI to identify weak areas in the power-generating system; PA1 d) determining actual safety margins that exist for these weak areas before collapse will occur; and PA1 e) shedding power in those portions of the power-generating system that are identified as weak areas, in order to prevent voltage collapse in the power-generating system, in accordance with said safety margins. PA1 1) It locally removes the singularity of the corresponding Jacobian. PA1 2) It requires only a simple modification of the original non-linear equations, with no added dimension. PA1 3) It adds just a few non-zero elements to the Jacobian matrix of large sparse systems. PA1 4) It enlarges the region of convergence around singular solutions. PA1 5) It leads to a new characteristic equation that defines the saddle-node bifurcation point by adding just one equation to the original set of non-linear equations.
The aforementioned method also contemplates, as an alternative to step "e", the step of adding power-generating capacity to weak areas, in order to increase the safety margin, and thus prevent voltage collapse during heavy load conditions.
Previous prediction indices could not determine where to shed load in the system. Neither could they determine or measure the load-sustaining capacities of the weak areas. Nor could they be used to pinpoint the underlying mechanisms of voltage collapse. In other words, such indices were developed in the "state space", not in the "parameter space". They were not directly correlated to load demands.
The invention reflects the observation that saddle-node bifurcations, which occur in physical applications such as power generation, form critical points, loading limits, extremes of parameter variation and points of loss of stability. In prior mathematical determinations of criticality, the singularity of the system Jacobian matrices at bifurcation points were usually determined by using continuation methods, as well as an extended characteristic equation of dimension 2n+1 for n-dimensional systems. The invention presents a mathematical method for a quick procedure that eliminates the convergence problems associated with the singularity of the Jacobian.
Thereafter, the invention defines: "decoupled, parameter-dependent, non-linear (DPDN) dynamic systems as ones whose dynamics can be represented by a set of non-linear equations with a varying parameter that can be decoupled from the remainder of the equation".
The invention further restricts the procedure to linear parameter-dependence.
The present invention first solves the convergence problem at or near the saddle-node bifurcation points (SNBP), and then estimates the location of these points in order to compute the exact SNBPs.
In order to perform the exact computation, however, the invention has developed a new method that reduces the computational burden of dealing with a (2n+1)-dimensional system of equations. The invention expresses a new characteristic equation defining the exact saddle-node bifurcation point, and has a dimension of only "n+1". In addition, the invention derives a criterion that allows for the bracketing of the solution in advance, in order to guarantee convergence. Such assurance has usually been impossible when using standard (2n+1)-dimensional, non-linear equations.